right adjoint

A functor $F : \C \to \D$ is a right adjoint when there is a functor $G : \D \to \C$ such that there are natural bijections $\Hom(G(A),B) \cong \Hom(A,F(B))$ .

Dual property: left adjoint
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Relevant implications

continuous implies right adjoint *
equivalence implies continuous andmonadic andright adjoint
monadic implies conservative andfaithful andright adjoint
right adjoint implies continuous

*Those implications also require assumptions on the source or target category.

Examples

There are 3 functors with this property.

contravariant power set functor
forgetful functor for vector spaces
identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

abelianization functor for groups
covariant power set functor
free group functor

Unknown

There are 0 functors for which the database has no information on whether they satisfy this property.

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